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Sep 23, 2013
09/13

by
Jyoti Champanerkar; Denis Blackmore

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A pitchfork bifurcation of an $(m-1)$-dimensional invariant submanifold of a dynamical system in $\mathbb{R}^m$ is defined analogous to that in $\mathbb{R}$. Sufficient conditions for such a bifurcation to occur are stated and existence of the bifurcated manifolds is proved under the stated hypotheses. For discrete dynamical systems, the existence of locally attracting manifolds $M_+$ and $M_-$, after the bifurcation has taken place is proved by constructing a diffeomorphism of the unstable...

Source: http://arxiv.org/abs/math/0411467v1

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17

Sep 21, 2013
09/13

by
Anatoliy K. Prykarpatsky; Denis Blackmore

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If we are given a smooth differential operator in the variable $x\in {\mathbb R}/2\pi {\mathbb Z},$ its normal form, as is well known, is the simplest form obtainable by means of the $\mbox{Diff}(S^1)$-group action on the space of all such operators. A versal deformation of this operator is a normal form for some parametric infinitesimal family including the operator. Our study is devoted to analysis of versal deformations of a Dirac type differential operator using the theory of induced...

Source: http://arxiv.org/abs/math/9907211v1

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27

Jul 22, 2013
07/13

by
Lu Ting; Omar Knio; Denis Blackmore

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Grobli (1877) laid the foundation for the analysis of the motion of three point vortices in a plane by deriving governing equations for triangular configurations of the vortices. Synge (1949) took this formulation one step further to that of a similar triangle of unit perimeter, via trilinear coordinates. The final reduced problem is governed by an integrable two-dimensional system of differential equations with solutions represented as planar trajectories. Another key to Synge's analysis was...

Source: http://arxiv.org/abs/0807.0454v1

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12

Jul 20, 2013
07/13

by
Denis Blackmore; Lu Ting; Omar Knio

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It is well known that the dynamics of three point vortices moving in an ideal fluid in the plane can be expressed in Hamiltonian form, where the resulting equations of motion are completely integrable in the sense of Liouville and Arnold. The focus of this investigation is on the persistence of regular behavior (especially periodic motion) associated to completely integrable systems for certain (admissible) kinds of Hamiltonian perturbations of the three vortex system in a plane. After a brief...

Source: http://arxiv.org/abs/math/0609427v1

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8.0

Sep 19, 2013
09/13

by
Denis Blackmore; Yarema A. Prykarpatsky; Roman Samulyak

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We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E. Cartan. Especially, the E. Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures on jet-manifolds and Cartan-Ehresmann connection theory on fibered spaces. General structure of integrable one-forms augmenting the two-forms associated with a closed ideal in the Grassmann...

Source: http://arxiv.org/abs/math-ph/9801203v1

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12

Sep 23, 2013
09/13

by
Denis Blackmore; Roman Samulyak; Anthony Rosato

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A new class of integro-partial differential equation models is derived for the prediction of granular flow dynamics. These models are obtained using a novel limiting averaging method (inspired by techniques employed in the derivation of infinite-dimensional dynamical systems models) on the Newtonian equations of motion of a many-particle system incorporating widely used inelastic particle-particle force formulas. By using Taylor series expansions, these models can be approximated by a system of...

Source: http://arxiv.org/abs/nlin/9904201v1

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27

Sep 23, 2013
09/13

by
Denis Blackmore; Morten Brons; Arnaud Goullet

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A simple - yet plausible - model for B-type vortex breakdown flows is postulated; one that is based on the immersion of a pair of slender coaxial vortex rings in a swirling flow of an ideal fluid rotating around the axis of symmetry of the rings. It is shown that this model exhibits in the advection of passive fluid particles (kinematics) just about all of the characteristics that have been observed in what is now a substantial body of published research on the phenomenon of vortex breakdown....

Source: http://arxiv.org/abs/0801.2408v1

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17

Sep 21, 2013
09/13

by
Nikolai N. Bogolubov Jr.; Denis Blackmore; Anatoliy K. Prykarpatsky

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We review the modern classical electrodynamics problems and present the related main fundamental principles characterizing the electrodynamical vacuum-field structure. We analyze the models of the vacuum field medium and charged point particle dynamics using the developed field theory concepts. There is also described a new approach to the classical Maxwell theory based on the derived and newly interpreted basic equations making use of the vacuum field theory approach. In particular, there are...

Source: http://arxiv.org/abs/1204.5129v4

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Sep 23, 2013
09/13

by
Yarema A. Prykarpatsky; Denis Blackmore; Jolanta Golenia; Anatoliy K. Prykarpatsky

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An approach based on the spectral and Lie - algebraic techniques for constructing vertex operator representation for solutions to a Riemann type Gurevicz-Zybin hydrodynamical hierarchy is devised. A functional representation generating an infinite hirerachy of dispersive Lax type integrable flows is obtaned.

Source: http://arxiv.org/abs/1105.4403v3

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10.0

Sep 22, 2013
09/13

by
Denis Blackmore; Jolanta Golenia; Yarema A. Prykarpatsky; Anatoliy K. Prykarpatsky

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Invariant ergodic measures for generalized Boole type transformations are studied using an invariant quasi-measure generating function approach based on special solutions to the Frobenius--Perron operator. New two-dimensional Boole type transformations are introduced, and their invariant measures and ergodicity properties are analyzed.

Source: http://arxiv.org/abs/1210.1746v3

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129

Jul 19, 2013
07/13

by
Bohdan Yu. Kyshakevych; Anatoliy K. Prykarpatsky; Denis Blackmore; Ivan P. Tverdokhlib

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A competing market model with a polyvariant profit function that assumes "zeitnot" stock behavior of clients is formulated within the banking portfolio medium and then analyzed from the perspective of devising optimal strategies. An associated Markov process method for finding an optimal choice strategy for monovariant and bivariant profit functions is developed. Under certain conditions on the bank "promotional" parameter with respect to the "fee" for a missed...

Source: http://arxiv.org/abs/1005.2661v1

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8.0

Sep 21, 2013
09/13

by
Denis Blackmore; Yarema A. Prykarpatsky; Orest D. Artemowych; Anatoliy K. Prykarpatsky

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The complete integrability of a generalized Riemann type hydrodynamic system is studied by means of symplectic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, Lax type representation and related infinite hierarchy of conservation laws are constructed.

Source: http://arxiv.org/abs/1204.0251v4